2003 A 3D host-immune-tumor system

Recent advances in oncology lead to promote the role of the tumor micro-environment in tumor growth [1] . In fact, mathematical cancer model taking into account normal (healthy) cells - thus the micro-environment - interacting with immune and tumor cells are not numerous. There is one proposed by Owen and Sherratt [2] which remains mainly focused on tumor–macrophage interactions, the normal cells being only considered for their ability to colonize the site studied. One of the most interesting models seems to be the model proposed by Lisette De Pillis and Ami Radunskaya [3] which
incorporates host,immune and tumor cells to reproduce certain qualitative aspects as oscillations in tumor size(Jeff’s phenomenon) [4] or tumor dormancy [5]. This model is rather generic in the sense that it is not specific to a given type of cancer. It is indeed based on quite common interactions between host, immune and tumor cells.The model is

where x designates the normalized population of host cells, y being the
population of effector immune cells and z the population of tumor cells. Tumor cells compete for resources with the two other populations of cells. From that point of view, they are the ``generalist’’ competitors while the two others are ``specific competitors’’.

This model produces a chaotic attractor for the set of parameter values as
follows [6]